Complex Analysis A Study Guide for the Midterm Exam Spring 2020
Complex expressions
1. Remember the polar form z = reiθ. 2. Let Σ := {(u, v, w) | u2 + v2+ (w − 1
2)2 = 1
4}. Stereographic projection ϕ : Σ \ N → C, where N = (0, 0, 1), is given by
ϕ(u, v, w) =
u
1 − w, v 1 − w
; ϕ−1(x, y) =
x
x2+ y2+ 1, y
x2+ y2+ 1, x2+ y2 x2 + y2+ 1
.
3. cos z = eiz+ e−iz
2 and sin z = eiz − e−iz
2i for z ∈ C.
4. log z = log r + iθ + i(2πn).
Holomorphic functions
1. To show that a function f is holomorphic, use the Cauchy-Riemann equations and show that the partials are continuous. You can also any rule from real analysis (the chain rule, the product rule, the quotient rule, etc.), or show that f has a power series development.
2. Let f (x, y) = u(x, y) + iv(x, y) := (u(x, y), v(x, y)). Recall that the Cauchy-Riemann equa- tions are
fy(z) = ifx(z) ⇐⇒ ux = vy, and uy = −vx.
3. f is holomorphic on an open set ⇐⇒ f complex analytic ⇐⇒ f smooth; this means you can substitute any holomorphic function with its power series, and can differentiate it endlessly.
4. The closed curve theorem tells us that the integral of a function that is holomorphic in an open disk (or an open polygonally simply connected region) D over a closed contour C ⊂ D is 0.
5. The antiderivative theorem, if f (z) is holomorphic on an open disk (or an open polygo- nally simply connected region) D, then it has an antiderivative F (z) such that F0(z) = f (z).
6. Cauchys integral formula allows us to determine the value of a holomorphic function f at a point. In particular,
f (w) = 1 2πi
Z
C
f (z) z − wdz,
where f is holomorphic in an open disk (or an open polygonally simply connected region) D and w is any point in the region bounded by the positively oriented C ⊂ D.
7. The uniqueness theorem tells us that holomorphic functions are determined uniquely up to their values in a certain region.
8. The mean value theorem allows us to evaluate a holomorphic function at point. If f (z) is holomorphic on D = D(p, R), then for any 0 < r < R, we have
f (p) = 1 2π
Z 2π 0
f (p + reiθ) dθ.
9. The maximum modulus principle says that if f is holomorphic in any region, then its maximum modulus is achieved at the boundary.
Complex Analysis A Study Guide for the Midterm Exam(Continued) Spring 2020 10. The minimum modulus principle says that if f is a non-constant analytic function in a region D, then no point z ∈ D can be a relative minimum of f unless f (z) = 0. For example if f (z) = z for z ∈ D = D(0; r), then min
D |f | = |f (0)| = 0, the mimimum modulus is 0 and it is achieved at an interior point.
Entire functions
1. Entire functions are functions that are holomorphic in all of C.
2. Liouvilles theorem tells us that any bounded entire function is constant. A generalization is that if |f (z)| ≤ A + B|z|N and f (z) is entire, then f (z) is a polynomial of degree ≤ N.
3. likewise, , if f (z) is entire and lim
z→∞f (z) = ∞, then f (z) is a polynomial.
4. The fundamental theorem of algebra says that a nonconstant polynomial always has a root.
Others
1. Know the denitions of radius of convergence, limk→∞|ck|1/k = L, and convergence tests/arguments for power series. These may be things that range from proving that a power series of a holomorphic function is uniformly convergent to finding the radii of convergence of some functions.
2. In evaluating a nontrivial integral, you need to know how to the closed curve theorem to give a contour integral as zero, then decompose the contour and bound the individual parts to find a specic integral.
3. A helpful tool is that you can bound the absolute value of any integral over CR, circle of radius R centered at 0, by 2πR times the integrand; this is known as the M L theorem or the estimation lemma.
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